How to design a shoe cap
Bridge Edge Loops is Blender's lofting tool, with a reasonable range of options.
One way to capture the curvature:


Rotate a duplicate of the top edge of your sole-rim upwards, towards the top profile of the shoe. If you have an actual front elevation of the shoe to work to, for the profile  edge, so much the better. (You could create a curve for that, convert to mesh, adjust the number of vertices to match the base edge. Loop Tools &gt; Space is very handy for getting a fixed count of vertices evenly spread along a curved mesh-loop.)
Either use an existing top-surface, or extrude a temporary approximate one out, to give Bridge Edge Loops something to work to, and then use Bridge Edge Loops, with 'Blend surfaces' set. Adjust the number of 'Cuts', and 'Smoothness' to get as close as possible to your desired result.
Delete the temporary surface, and maybe Relax or Smooth some regions of vertices to improve the shape.

All this happens before detailing, and subdivision.